What is Huygens–Fresnel principle? — Physics

Definition 22.1 University Physics — Year 2 · Chapter 22 — Fraunhofer Diffraction and Spatial Filtering

Every point of a wavefront (or of an aperture lit by a wave) acts as a secondary source emitting a spherical wavelet, in phase with the wave there; the wave beyond is the coherent sum of the wavelets. For an aperture $\Sigma$ lit by a plane wave of amplitude $a$ , the amplitude reaching a far point is $\underline s(M) = K\iint_\Sigma\eu^{-\iu\varphi(P, M)}\dd S$ , with $\varphi(P, M)$ the phase of the path from the aperture point $P$ to $M$ ( $K$ a constant). Fraunhofer (far-field) diffraction is the case where $M$ is at infinity — or in the focal plane of a lens — so that the rays from all points of the aperture toward $M$ are parallel, in the direction $\vect u$ , and

\varphi(P, M) = \varphi_0 - \frac{2\pi}\lambda\,\vect{OP}\cdot\vect u , \qquad \underline s(\vect u) \propto \iint_\Sigma\eu^{\iu(2\pi/\lambda)\vect{OP}\cdot\vect u}\dd S .

The pattern is observed at infinity, or at the focus $F'$ of a lens, where the direction $\vect u$ of small angles $(\alpha, \beta)$ maps to the point $(f\alpha, f\beta)$ .

Examples

Example 22.5 (Eye, telescope, microscope)

The eye ( $D = 3\,\mathrm{mm}$ , $\lambda = 550\,\mathrm{nm}$ ): $1.22\lambda/D = 2.2 \times 10^{-4}\,\mathrm{rad}$ , about $45''$ — a $1\,\mathrm{mm}$ detail at $4\,\mathrm{m}$ , which is indeed about the acuity of a good eye: evolution matched the retina’s cells to the diffraction limit. A $2.4\,\mathrm{m}$ telescope: $0.06''$ , a thousand times better, if the atmosphere lets it (it does not, from the ground: $1''$ of "seeing", whence space telescopes and adaptive optics). A radio dish of $100\,\mathrm{m}$ at $\lambda = 21\,\mathrm{cm}$ : $9'$ , worse than the eye; interferometers across continents restore the resolution. A microscope objective of aperture angle $\alpha$ resolves $0.61\lambda/n\sin\alpha$ — about $0.2\,\text{µ}\mathrm{m}$ in the visible, whatever the magnification: the limit that drove microscopy to electrons and to X-rays.

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Examples

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